"An Introduction to Mathematical Reasoning" by Peter J Eccels. Teaches the vocabulary of mathematics, just the basics you need to think like a mathematician, not a mathematics user like most science texts.

"How To Solve It" George Polya. Heurists and problem solving skills, by a great mathematician.

Do a google search, specially in the sci.math newsgroup. Again, read books by mathematicians for mathematicians; they're often far more enjoyable and actually far more straightforward (I was often confused by the examples in my school work; I didn't care for "vehicle moving at speed X" or "object falling at from height Y". We all have a different internal visual mind and I tended to think in abstract patterns, usually colors, lines or nested bodies, without real physical objects distracting me.)

How To Solve It is a great recommendation.

I had a physics professor that did an amazing job of teaching problem solving; even though the class wasn't directly to my major, it was probably the single most useful class I ever took. Next semester I read How To Solve It and realized that the way he taught followed closely to what Polya lays out.

Fantastic reply, thank you.

An Introduction to Mathematical Reasoning loks good, just going to read some reviews on Amazon first.

Would 'How To Solve It' be aimed at someone like myself?

How to Solve It seems to be aimed at math teachers, but it is useful for almost anyone wanting to learn more math since it gives ideas about how to think about maths.

I second the Eccles recommendation. It's a really great introduction to formal mathematics.

What kind of math do you want to learn?

There's a lot of great resources out there, but you need to be more specific.

For example: I really enjoyed Gilbert Strang's course on Linear Algebra, available as a series of video lectures on MIT OCW (http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/Cours...).

If you are interested in Discrete mathematics, Knuth's "Concrete Mathematics" is a great book--but it might not count as "basic" enough for your purposes, depending on your background.

If by "basics" you mean "the stuff you should have learned in high school or as an undergrad", the Standard Deviants videos are fun: http://www.sdlearn.com/default.asp

Those Strang videos demonstrate that there are benefits to lecturing in front of an empty chalkboard (or several, in his case), chalk in hand, vs. displaying slides from a computer. Watching how he steps back and thinks out loud about what he just wrote and what to do next, double checking his work, making and correcting the occasional error, all greatly enhance the pedagogic impact.

Concrete Mathematics is (as it says itself) intended for first year grad students. It is certainly good, but it is NOT for a "complete noobie". I am in a similar situation, and am finding Vellman's "How to Prove It" extremely useful. Having poked around the edges of math for about a year, I finally realized that proofs are central to mathematical maturity... No more eyes crossing when I read a new CS textbook!

Basically high school stuff and up

Here is some general advice.

First have some kind of goal. Do you want to be able to determine the orbit of an planet, evaluate the complexity of algorithms and computing models, study human interactions as walks on graphs, or use statistics to model and predict complex systems. Decide on this first, don't wonder around mathematics aimlessly.

Secondly, work the books. Maths can not be learned by observation, and reading proof after proof is simply observation. Memorize proofs, work from your current point back to first principles, and do all the problems you can. Of course there will be times when you simply can not find a means to start on a problem, and at that time find help or try and come back to the problem later.

I'm in the similar situation as you (OP). My high school education was interrupted quite badly and 13 years after graduating I lack confidence in my comp sci endeavors because my maths sucks so bad. I'd be interested to hear of any hackers who have missed education milestones (like high school maths) but gone and successfully filled in the gaps. The reason I'm asking is because I'm kind feeling that things like maths knowledge is layered on year after year and if you lack the foundation its really a huge amount of work to repair each successive layer.

Passing on some wisdoms to the young hackers around here... I wish someone had grabbed me by the face in highschool and told how important all these layers of skills/knowledge would be for getting the kind of jobs I want now. <I come from a blue collar background - by the time I realised how important education was (age 22) it was too late to do much about it>

I left school when I was 16 to do homeschool, which turned out to be nothing more than my parents buying me books when I asked for them and nothing more.

I did my GED and dropped out of college, so I think that I can relate to how you feel. There's a huge inferiority complex that comes from having less education than others. Filling in the gaps is difficult but I've found that whenever I fill something in, I've benefited pretty quickly. Lately I realized that I sucked at systems programming concepts, so I've been reading about that, which has been really helpful. I make a living as a programmer, but I have to continually be studying to try and fill in gaps before I'm hurt by lack of knowledge.

If someone can work through a discrete math textbook and do the exercises, I think they could get a lot out of it. Calculus has some important ideas, but I think the discrete math is much more relevant to every day programming. I took a course last year at my community college that used Discrete Mathematics with Applications, by Susanna S. Epp. I don't recall it requiring higher math than algebra, but it did require sharp thinking. Concrete Mathematics: A Foundation for Computer Science is probably a good textbook, but I haven't used it, so can't say anything about it.

"The reason I'm asking is because I'm kind feeling that things like maths knowledge is layered on year after year and if you lack the foundation its really a huge amount of work to repair each successive layer."

Absolutely, the layers you mention is exactly why I want to start from scratch

I find the Khan Academy videos to be pretty helpful. They start with the absolute basics and go on up. http://www.khanacademy.org/

Thank you

"What is Mathematics" by Courant and Robbins is quite good and respected, but it will challenge you: http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-... It may be more advanced then what you're looking for though.

Have you seen this thread?

http://news.ycombinator.com/item?id=665029

That thread recommends many very few good books, but probably mostly books too hard at first for the participant who has posted this new thread.

I'll recommend a couple of books from that thread:

http://www.springer.com/physics/book/978-0-306-45036-5

http://www.amazon.com/Mathematics-Short-Introduction-Timothy...

I agree with the recommendation of An Introduction to Mathematical Reasoning in this thread.

Another participant has already recommended my favorite for background reading, Concepts of Modern Mathematics by Ian Stewart.

http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewar...

Get that right away.

Sawyer's A Mathematician's Delight is surely also good (I've read other books by Sawyer).

http://www.amazon.co.uk/gp/product/0486462404/

Read those for background as you get my favorite overviews of mathematics: Basic Mathematics by Serge Lang and Numbers and Geometry by Joseph Stillwell.

http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/038796...

(Basic Mathematics is mostly high school level math, with a minimum of fuss and bother, and good exercises.)

http://www.amazon.com/Numbers-Geometry-John-Stillwell/dp/038...

(Numbers and Geometry is mostly undergraduate level math, with very good explanations and excellent exercises.)

I've had pretty good luck using the MAA recommendations list for libraries: http://mathdl.maa.org/mathDL/19/?pa=content&sa=viewDocum...

I recommend reading Theodore Gray and Jerry Glynn's Brain Rot article for some ideas on what math skills are worth intensive study and development and which are less important: http://www.theodoregray.com/BrainRot/

There are several articles and blog postings on the topic of math self study that I found interesting and you might find useful in determining what and how to study: Developing your intuition for math: http://betterexplained.com/articles/developing-your-intuitio... Math every day (Steve Yegge): http://steve.yegge.googlepages.com/math-every-day Math for programmers (Steve Yegge): http://steve-yegge.blogspot.com/2006/03/math-for-programmers... How to read mathematics (Shai Simonson and Fernando Gouvea):http://web.stonehill.edu/compsci/History_Math/math-read.htm

If you go back and learn Algebra, Trig, Geometry, then I fully recommend the Cliffs Study Solver series of textbooks because they are very cheap and very thorough plus each day you do a chapter, you'll make cumulative progress.

You are introduced to a concept, given a set of practice problems to see the concept in action, then given a problem set to solve on your own.

http://www.amazon.com/Algebra-I-Cliffs-Study-Solver/dp/07645...

Each book is about 350 pages and you'll be up to speed in no time.

I came back to doing maths after a gap of over a decade. I found the student survival guide very clear and useful. I imagine it would be excellent to someone who is not good at maths.

I read the guide every evening while I was cooking (its not a difficult read) and my maths improved greatly.

http://www.netcomuk.co.uk/~jenolive/ http://www.amazon.com/Maths-Students-Survival-Self-Help-Engi...

That's depends on what you want to learn and for why, well, some people want to understand the formalism of a theory, as theoretical computer science or theoretical physics where others are only interested in applications, so I will take a generalist approach in the topics, yes topics not books, that I will advise you to learn. Unless the book is awful (and there are many out there that are) it will makes no difference which book you pick, you generally will not "read" a math book, the only case in which you will is when it's a book for divulgation (as Polya's "How To Solve It"). For me the basics is: Statistics (Descriptive and some Probability), Calculus, the idea of Limits, Derivatives and Integrals (for Multiple Variables) and applications, Linear Algebra and also some Applications (there are many), numerical Linear Algebra is totally necessary if you want to apply it in the real world, a basics in Differential Equations, some Numerical Analysis.

If you want to learn things closer to computer science then learn something of Number Theory, some Enumerative Combinatorics and Graph Theory as well. The list is extensive because I come from a mathematical background. If you learn at least a bit of these topics them the next step will be apparent for you.

I know this doesn't answer your question since you asked about math in general, but in case anyone ever starts a "Best Physics books for complete noobie?" thread I'd like to go ahead and suggest Brian Greene's "Fabric of the Cosmos": http://www.amazon.com/Fabric-Cosmos-Space-Texture-Reality/dp...

Maybe more on topic than you suggest, as some Physics books double as pretty good books about math.

http://www.amazon.com/Road-Reality-Complete-Guide-Universe/d...

OK, I haven't actually read it, but it looked like a really good book for learning a lot of interesting math when I thumbed through its contents at the library. :)

This is not a good book for someone who doesn't know math. He's on to hyperbolic geometry by ch 2. (unsuccessfully) reading this book was one of the things that convinced me to back to school to study more math :-)

I agree with hyperbovine -- Road To Reality is a downright terrible book for the casually-interested nonprofessional, and I don't understand why it gets recommended so frequently. Roger Penrose is a bright fellow and a good writer but this book is not for the person who did OK in high school math and physics and now wants to take it to the next level.

How would it be for a guy who aced second semester freshman physics and the math GRE (40 years ago), and top 100 on the Putnam?

I'd guess you'd be in his target audience. Someone who's already comfortable with complex causality and relationships between seemingly-random facts.

Honestly I think Schaum's is pretty good because it does a bit of explanation, but you primarily learn through doing two dozen pages of problems per chapter.

Some years ago I took the same approach to start again from ground zero. I found Polya to be good, but a Mathematician's Delight is better and more accessible: http://www.amazon.co.uk/gp/product/0486462404?ie=UTF8&ta...

If you want to DO math, nothing beats working through a decent textbook. I have worked through several since I'm approaching 50 and don't use math enough to keep my skills up (and like any skill, you have to keep practicing to stay decently competent in math). The best Precalculus textbook I have used is Swokowski's "Algebra and Trigonometry with Analytic Geometry" which is clear, concise, and has lots of problems.

If you want to learn ABOUT math, Davis and Hersch's "The Mathematical Experience" is a fairly easy read about philosophy of math, how it is used, and a bit about studying math.

Eric Temple Bell's "Mathematics: Queen and Servant of Science" is a bit dated but an excellent history of math for someone interested in learning to do advanced math; the author's a bit biased towards algebra over analysis, number theory, and geometry, but not excessively so. Its biggest lack is only one short chapter on probability and statistics. This is not a particularly easy read since it covers things in some depth, but I think it is worth the effort.

I haven't picked it up yet, but I remember reading about "The Princeton Companion to Mathematics" on here a while ago. It looks like a pretty complete guide to all of modern mathematics and sounded like it was easy enough for a beginner to get through while still being able to teach math experts some new things.

Thanks for reminding me about it - I think I'm going to order my self a copy!

My own math background is through Calculus II right now, so I'm a beginner, but I can't say the Companion (though it is excellent) is something I could 'get through' in the sense that people usually mean it. By necessity it is terse, despite its massive size. I personally find it useful as a survey on the field, but I wouldn't recommend it as a teaching guide in and of itself for people who happen to be as dumb as me.

For proof based mathematics, I found "How to Prove It:A Structured Approach" helpful.

I heartily endorse reading the classic works of geometry as a way to both the subject as well as a way of thinking about math, proof, and argumentation.

Start with Euclid's _Elements_, and then move onto Archimedes' short books on levers and floating bodies, Apollonius's wonderful treatise on conic sections, and Ptolemy's _Almagest_.

They are excellent for self-education, providing both geometrical knowledge in itself, as well as extensive training in sound reasoning. Don't be fooled by the antiquity of their origin: they teach more clearly than most modern day texts, and their content is timeless.

I completely disagree. Do not rely on ancient texts to give you understanding of what mathematics is about. They're very difficult to read, they do not teach well, and you'll know next to nothing about even basic notions of mathematics, as understood now, after you're through with them.

Do read the classical texts for pleasure, to round out your education, or to understand the history of mathematics. But don't study math from them, that'd be a terrible mistake.

I'd be curious to know what you find difficult about them, and why you think they do not teach well.

I find Euclid extraordinarily clear in his exposition of geometry. And I have not yet found a better teacher on conics than Appolonius, although Descartes comes very close with his analytic geometry.

Are there some defective constructions in the Conic Sections or in the Elements that lead you to say that "you'll know next to nothing about even basic notions of mathematics"?

It's not clear to me why you think that the ancient texts are lacking in pedagogical power, except for perhaps a personal aesthetic preference.

I will join with anatoly in disagreeing. The Elements are a beautiful piece of literature and I recommend reading them for a historical understanding. (I have not read them through by any means, but I have read segments which are magnificent).

As for understanding mathematics and its techniques, there are now much better methods. Our understanding of mathematics and methods of learning (not to mention our simple culture in general) have changed substantially since then, the most obvious point being the development of non-Euclidean geometry.

To add to the list of books already mentioned consider Numeracy. It was influential for me when I was an undergraduate and touches on numerous topics. Also, if you want to start at the beginniner, there is a very good book called Chapter Zero that starts with nothing and is easily accessible to anyone willing to study it.

You say that "our understanding of mathematics and methods of learning have changed substantially since then".

While I of course agree that we now have a lot more mathematics, including non-Euclidean geometry, I don't see what thay has to do with the value of Euclid and the other ancient mathematicians. I'm not recommending Euclid and Appolonius as a complete mathematical education, but a good place to start.

What are the methods of learning that we now have that invalidates Euclid and the others from being valuable teachers of mathematics?

Invalidates? Absolutely none. Good mathematics is eternal and the Elements has some beautiful mathematics.

What I meant to say is that in the fullness of time they have been expanded on and surpassed. You can still learn good and valid mathematics from Euclid, but you can learn it better form more modern sources.

As one example amoung many, good modern geometry books will let the reader know about the existence of non-Euclidean geometry and its differences, even if the focus is on traditional Euclidean geometry.

Also, modern books will use modern notation. Notation of course does not affect the ideas which are what is important, but someone trying to learn math today will need to learn the notation so that they can read other more advanced books that will assume the reader knows that stuff. Not to mention, notation can make things easier or harder (try doing multiplication in Roman Numerals).

Finally, many things that were proven tediously before have easier proofs developped over time, often with the ability to take recourse in Algebra. For the same reason, many open questions in Euclid's time now have answers.

What you say makes a lot of sense.

However, notation actually does affect the ideas, and deeply. Euclid did not consider 1 a number, for perfectly sound reasons. Multiplication is a very different animal in Euclid's math, and his proportions are not just funny-looking fractions.

Overall, you are right. If someone is looking to get a broad survey of mathematics, Euclid is not a good source. But to start from scratch and learn rigorous math-by-proof, he has no equal.

I'm pretty at Mathematics and doing that takes a lot of work. And the Mathematics books for noobs are no good.

The better approach is to get a theoretical book, something like Spivak's Calculus or Linear Algebra by Friedberg, Insel and Spence. And then from there whenever you have difficulty with the material spread out laterally and you really start to gradually grow an understanding of mathematics.

And then perhaps, one day you'll be up for Spivak's Calculus on Manifolds!

The World of Mathematics edited by James Newman (a four-volume anthology) may give you a feel for and an interest in exploring math further. It's a great collection ranging from mathematical curiousities and puzzles to memoirs of mathematicians to a very moving short story by Aldous Huxley.

Also, One, Two, Three, Infinity by George Gamow (a great physicist) is a great intro.

Second the How To Solve It recommendations.

If you have some decent background in reasoning and logic, I would suggest Linear Algebra Done Right by Sheldon Axler. It provides a general introduction in mathematical reasoning as well as providing a strong framework for maths needed in many fields.

http://www.amazon.com/Linear-Algebra-Right-Sheldon-Axler/dp/...

For a "popular" treatment of mathematics that does go into some mathematical detail "Journey through Genius: The Great Theorems of Mathematics" by William Dunham is difficult to beat. It is by far one of the best "Math" books I have read that have kept me coming back to it. Also, try some of the books by John Derbyshire along similar lines.

I second the OCW reference. Some of the Calculus for Dummies, type books are good. Its good to remember that Calculus and Linear Algebra don't have to be that complicated. I also recommend scan the books before you buy them, I wasted far too much money at college on txtbooks that I ended up despising

It's not strictly math, and I can't recommend it from experience, but I've always been curious what a beginner's reaction to _Structure and Interpretation of Classical Mechanics_ might be. It's free here:

http://mitpress.mit.edu/SICM/book.html

"Scientific Notation and Other First Principles: Comprehensive Mathematics for Lawyers and Politicians," by Jacob Herwitz. Penguin, 1992.

Great introductory text which starts from algebraic first principles and goes through pretty much everything up until differential equations. Very thorough.

I think Kreyszig's Advanced Engineering Mathematics is an excellent text in applied math.

I'm not sure where you're at or what direction you want to study, but once you're familiar with calculus concepts this text is a good place to go to deal with ODE and analysis.

Concepts of Modern Mathematics

http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewar...

There are a few math books in the "Head First" series (which I always love). Those might serve more as refreshers for most people, but for a true math noob (like me), they have really helped.

"Naive Set Theory" by Paul R. Halmos.

Set Theory is the building blocks of mathematics, and Halmos is a first-rate mathematician and teacher. Excellent recommendation Paul. I worked through Halmos 8 months into my mathematics self-study and loved it.

'Zero: The Biography of a Dangerous Idea' by Charles Seif